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the second of which is to be applied to the x in the first. We get

qS)z -r W -r qS)

~~    (rP -r sR)z -r (rQ - sS) '

Attending only to the coefficients in the three transformations
we write them in square arrays, thus


and see that the result of performing the first two transforma-
tions successively could have been written down by the follow-
ing rule of 'multiplication'.


" \\R   SJJ      jjrP + sR    rQ + sS

where the rows of the array on the right are obtained, in an
obvious way, by applying the roars of the first array on the left
onto the columns of the second. Such arrays (of any number of
rows and columns) are called matrices. Their algebra follows
from a few simple postulates, of which we need cite only the

following. The matrices

are equal (by


*[|"[IC   D\
definition) when} and only when, a = A, b = B, c = C,d = D.
The sum of the two matrices just written is the matrix

a     b

C   d-t

The result of multiplying
i   mb

by m

c     d

The rule for 'multi-

(any number] is the matrix {(

ijmo      /rtiijj

plying', x, (or 'compounding') matrices is as exemplified for

p   q\     JiP   Q      ,

!     ;    ;   above,
r   Sj     {{12   5^

A distinctive feature of these rules is that multiplication is
not commutative^ except for special kinds of matrices. For
example, by the rule we get

3   _

Pq + Qs